AA (Angle-Angle) Similarity Concept Explanation:
The AA similarity theorem states that if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. When triangles are similar, their corresponding sides are proportional, meaning their side lengths share a common ratio.
This property helps solve problems involving the side lengths of triangles without requiring knowledge of all three angles or sides.
Diagram:
Let’s draw two triangles that demonstrate AA similarity:
*** QuickLaTeX cannot compile formula:
\begin{array}{c}
\begin{tikzpicture}
\draw (0,0) -- (3,0) -- (1.5,2) -- cycle;
\draw (5,0) -- (7,0) -- (6,1.5) -- cycle;
\draw (0,0) node[below] {A} -- (1.5,2) node[above] {B};
\draw (1.5,2) -- (3,0) node[below] {C};
\draw (5,0) node[below] {D} -- (6,1.5) node[above] {E};
\draw (6,1.5) -- (7,0) node[below] {F};
\draw (0.5,0.5) node {50^\circ};
\draw (2.5,0.5) node {80^\circ};
\draw (5.5,0.5) node {50^\circ};
\draw (6.5,0.5) node {80^\circ};
\end{tikzpicture}
\end{array}
*** Error message:
Missing $ inserted.
leading text: \begin{array}{c}
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leading text: \draw (0.5,0.5) node {50^
Extra }, or forgotten $.
leading text: \draw (0.5,0.5) node {50^\circ}
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leading text: \draw
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leading text: \end{tikzpicture}
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leading text: \end{tikzpicture}
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leading text: \end{tikzpicture}
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leading text: \end{tikzpicture}
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leading text: \end{tikzpicture}
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leading text: \end{tikzpicture}
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leading text: \end{array}
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leading text: \end{array}
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Hard Questions:
Q1: Triangle ABC has angles 40°, 60°, and 80°. Triangle DEF has angles 40°, 60°, and 80°. Are these triangles similar? Justify your answer.
Step-by-Step Solution:
Since two angles in triangle ABC are congruent to two angles in triangle DEF, the triangles are similar by the AA similarity theorem
.
Answer:
- Yes, the triangles are similar.
Q2: In triangle XYZ, the angles are 30°, 60°, and 90°. In triangle PQR, the angles are also 30°, 60°, and 90°. If side XY = 5 and side PQ = 15, what is the ratio of the corresponding sides?
Step-by-Step Solution:
The triangles are similar by the AA similarity theorem because their corresponding angles are equal. Therefore, the sides of these triangles are proportional:
- Ratio of corresponding sides = ( \frac{PQ}{XY} = \frac{15}{5} = 3 )
Answer:
- The ratio of corresponding sides is 3:1.